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Higher dimensional Chern-Simons supergravity Ma´ximo Banãdos1, Ricardo Troncoso1,2 and Jorge Zanelli1,3 1Centro de Estudios Cient´ıficos de Santiago, Casilla 16443, Santiago, Chile 2Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matemáticas, Universidad de Chile, Casilla 487-3 Santiago, Chile 3Departamento de F´ısica, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile ∇g =0. (2) A Chern-Simons action for supergravity in odd- A1...An+1 dimensional spacetimes is proposed. For all odd dimensions, [In the case of a supergroup, the invariant tensor should thelocalsymmetrygroupisanontrivialsupersymmetricex- have the corresponding (anti-) symmetry properties.] tension ofthePoincaré group. In2+1dimensionsthegauge For a given n, the above condition may have no so- group reduces to super-Poincaré, while for D=5 it is super- lutions. Indeed, (2) imposes strong restrictions on the Poincaré with a central charge. In general, the extension is group. As we shall see below, for D > 3, one needs to 6 obtained by the addition of a 1-form field which transforms enlarge the bosonic sector of the theory in order to pro- 9 asanantisymmetricfifth-ranktensorunderLorentzrotations. duce a supersymmetric extension of Chern-Simons grav- 9 Since the Lagrangian is a Chern-Simons density for the su- ity that contains local Poincaréinvariance. 1 pergroup,thesupersymmetryalgebra closes off shellwithout Itisdirecttoprovefrom(1)that,uptoatotalderiva- theneed of auxiliary fields. n tive, L is invariant under the gauge transformation a PACS numbers: 04.50.+h, 04.65.+e, 11.10.K J δλA=∇λ, (3) 3 I. INTRODUCTION where λ is an arbitrary zero-form parameter and ∇λ = 1 dλ+[A,λ] . If δ and δ are two trasformations, with v λ η In the search for an unified theory of all interactions parameters λ and η respectively, then 3 including gravity, higher dimensional models have be- 0 0 come standard in theoretical physics. Two important [δλ,δη]A=δ[λ,η]A. (4) 1 examples are D = 10 superstrings [1] and D = 11 su- 0 pergravity [2] which give rise, upon dimensional reduc- The three dimensional supergravity studied by 6 tion, to interesting effective models in four dimensions. Achućarro and Townsend [4], as well as the five- 9 In the study of higher –and lower– dimensional mod- dimensional theory studied by Chamseddine [5], are / c els, Chern-Simons densities play an important role. Al- Chern-Simons theories in the sense described above. q thoughChern-Simonsformsfirstappearedinthephysics Their supersymmetry transformations can be written in - literatureinthecontextofanomalies,itisnowclearthat the form (3) and therefore the supersymmetry algebra r g theyhaveanintrinsicvalueasdynamicaltheoriesintheir closes off-shell without the need of auxiliary fields. : own right. For example, pure gravity and extended su- Itgoeswithoutsayingthat,appartfromtheinvariance v i pergravityinthreedimensionsareChern-Simonstheories under (3), the Chern-Simons action is also invariant un- X for the groups SO(2,2) [3] and OSp(2|p)⊗OSp(2|q) [4], der diffeomorphisms. In 2+1 dimensions that symmetry r respectively. Similarly, in five dimensions, supergravity is not independent from the local gauge group because, a can be written as a Chern-Simons action for the super- as a consequence of the equations of motion, the con- groupSU(2,2|N)[5]. Chern-Simonsformsalsoprovidea nection is locally flat. This means that, given two con- simpledescriptionforpuregravityinallodd-dimensional figurations that differ by a diffeomorphism, there always spacetimes[6]. Theequationsofmotionofthesetheories exist a gauge transformation that deforms one into the possess black hole solutions [7] generalizing those found other. In the canonicalformalism this is reflected by the in 2+1 dimensions [8]. absence of new independent constraints associated with The Chern-Simons Lagrangian is constructed as fol- diffeomorphisms. lows. Let G a basis for the Lie algebra of a given (su- Indimensions greaterthan three,however,the Chern- A per)group G. Let A = AAG be the connection for G Simons equations of motion do not impose the flatness A and F = dA+A∧A = FAGA its curvature 2-form. The conditionand therefore the diffeomorphisminvarianceis Chern-Simons Lagrangian, L, is a (2n+1)-form whose anindependentsymmetrygivingrisetoindependentcon- exteriorderivative(definedin2n+2dimensions)satisfies: straints in the canonical formalism (see [9] for more de- tails on this point). For our purposes here it is enough dL=<F∧···∧F >, to observethat the gaugetrasformations(3) forma sub- =gA1...An+1FA1∧...∧FAn+1 (1) group of the whole symmetry group. It is the purpose of this paper to show that the above whereg ≡<G ,...,G >iscompletelysym- schemecanbeextendedtosupergravityinallodddimen- A1...An+1 A1 An+1 metric and satisfies the invariance condition sions,providedonechoosesthebosonicLagrangianinan 1 agprapnrgoiparniaftoerwthaey.boItsotnuircnssecotuotrtihsantotfoHriDlbe>rt’3s., RthaethLear-, δI =2Z ǫabcdRab∧Tcλd, (10) thecorrectLagrangianisnon-linearinthecurvatureand which is not zero for arbitrary λ. yet, gives rise to first order equations for the tetrad and An alternative approach, often followed in the super- the spin connection. gravity literature, is the so called 1.5 formalism. That is to set Ta = 0 keeping only the tetrad transformation in (7); the variation of wab is then calculated using the II. GENERAL RELATIVITY AS A GAUGE chain rule. This procedure brings the diffeomorphism THEORY invariance into the scene because if Ta = 0, the variation (7) for the tetrad is equal -up to a rotation- to Inthissectionweshallreviewsomeaspectsoftheviel- a Lie derivative with a parameter ξµ = eµλa. In sum, bein –or gauge– formulation of general relativity. The a the Hilbert action in 3+1 dimensions is invariant under main point of this section is to display the differences Lorentz rotations and diffeomorphisms, but not under between gravity in odd and even dimensions. the local translations (7), generated by P . a A completely different situationis observedin 2+1di- mensions,inwhichcasetheHilbertactionislinearinthe A. Poincaré translations vs. diffeomorphisms dreibein field, Generalrelativityinfour andallevendimensionscan- not be construed as a ‘truly’ localgauge theory [10]. On I2+1 =Z ǫabcRab∧ec. (11) thecontrary,forodd-dimensionalspacetimes,gravitycan be written as a Chern-Simons action for the Poincaré It is straighforward to check, using the Bianchi iden- group [4,3,6]. [If the cosmological constant is present, tity,thatthevariationof(11)under(7)givesaboundary then the relevant group is the (anti)-de Sitter group.] termandhence,(7)isasymmetryofthe2+1theory. As The Poincaré group has generators P and J satisfy- Witten has pointed out, this simple fact has deep conse- a ab ing: quences[3]. Indeed, inthreedimensions,onecanreplace thediffeomorphisminvariancebyalocalPoincaréinvari- [Pa,Pb]=0, ance whose constraint algebra is a true Lie algebra. [P ,J ]=η P −η P , Onemaywonderifthereexistsanactioninvariantun- a bc ab c ac b der (7) in higher dimensions. This generalization indeed [J ,J ]=η J −η J +η J −η J . (5) ab cd ac bd bc ad bd ac ad bc exist and is given by We define the connection for this group 1 I2n+1 =Z ǫa1...a2n+1Ra1a2∧···∧Ra2n−1a2n∧ea2n+1 (12) A=eaP + wabJ . (6) a ab 2 which, clearly, exists only in odd dimensions. The fact Letλbe anarbitraryparameterwithvaluesintheLie that (12) can be written for odd dimensional manifolds algebra (λ = λaP + 1λabJ ). Under the infinitesimal onlyisassociatedtotheexistenceofChern-Simonsforms a 2 ab gaugetransformationδA=∇λ,ea andwab transformas in those dimensions. The key property of (12) is that it is linear in the follows. vielbein field rather than being linear in the curvature, Translations: δea =Dλa, δwab =0, (7) as would be the case for the Hilbert action. This fact Rotations: δea =λaeb, δwab =−Dλab, (8) makes (12) invariant under the transformation (7), up b to a boundary term. Inspite of being non linear in the where D is the covariantderivative in the connection w. curvature, this action yields first order differential equations for all the fields. This is not surprising as (12) is a Itmightseempuzzlingthateventhoughthetetradand particularcase of a Lovelockaction[11]. I describes thespinconnectioncarryarepresentationofthePoincaré 2n+1 a Chern-Simons theory of ISO(D − 1,1), obtained by group,theHilbertactionconstructedpurelyoutofthose contractionofSO(D−1,2),andpossessessolutionswith fields, is not invariant under the Poincaré group in four conical singularities [7] analogous to those found in 2+1 dimensions. In fact, the action dimensionswithoutcosmologicalconstant[12]. Itshould be mentioned here that the action (12) has a propagat- I =Z ǫabcdRab∧ec∧ed (9) ing torsion [13] and therefore, the 1.5 formalism is not applicable in this case. The main goal of this paper is is invariant under (8) but not under (7). The reason is to describe the supersymmetric extension of the action simplythatunder(7)theactionchanges,modulobound- (12). ary terms, by 2 B. Super-Poincaré vs. supergravity to this case [4], there is an alternative route which leads more directly to supergravity. The key feature of 2+1 Supergravity if often referred to as the square root of supergravity is that –unlike 3+1 supergravity–, the ac- General Relativity [14] much in the same spirit as the tion is invariant under local Poincaré transformations. Dirac equation is the square root of the Klein-Gordon This means that in the 2+1 theory it is not necessaryto equation. This is justified by the fact that the commu- expressthe righthandside of(14)interms ofdiffeomor- tatoroftwosupersymmetrytransformationsgivesagen- phisms;the commutatoroftwosupersymmetrytransfor- eral change of coordinates plus a rotation and another mationsgivesalocaltranslation,whichis asymmetryof supersymmetry transformation. A concrete example is the action as well. the usual N = 1 local supersymmetry transformations In other words, in 2+1 dimensions one can consider a [15] truly first order formalism in which the spin connection transforms independently of the vielbein and gravitino 1 just as dictated by group theory. In particular, under δea = ε¯Γaψ , µ 2 µ supersymmetry, one can set δψ =D ε. (13) µ µ δwab =0 (16) µ Here ψ and ε are Majorana spinors; ε¯ is the Majorana conjugate (ε¯) = C εα satisfying ε¯Γaη = −η¯Γaε, and which, together with (13), is a symmetry of the 2+1 ac- D is the covαariantαdβerivative in the spin connection. It tion. µ The simplicity of the 2+1 theory can be explicitly ex- is then straightforwardto prove that the commutator of twotransformationswithparametersεandη actsonthe hibited. The action reads [4,16] tetrad as 1 I =Z (ǫabcRab∧ec−ψ¯∧Dψ), (17) [δ ,δ ]ea = D (ε¯Γaη). (14) ε η µ 2 µ where ψ is a two component Majorana spinor. [See the The transformation(13)followsfromthe super-Poincaré Appendix for a summary of conventions.] algebrawhenoneconsiderstheveilbeinandthegravitino This action is invariant under Lorentz rotations, asthecompensatingfieldsforlocaltranslations(P )and a supersymmetry transformations (Q), respectively. It is δωab =−Dλab, δea =λaea, b notsurprisingthereforethattherighthandsideof(14)is 1 alocaltranslation–actingonthetetrad–,withparameter δψ = λabΓabψ; (18) 4 λa = 1ε¯Γaη. 2 The trouble with supergravity in 3+1 dimensions is Poincarétranslations, that the action is not invariant under local translations. Nevertheless, transformation (14) is still a symmetry of δwab =0, δea =Dλa, δψ =0; (19) the action provided the transformation of the spin con- nectionpreservesthetorsionequationTa = 1ψ¯Γa∧ψ(1.5 and supersymmetry transformations, 2 formalism). On the surface defined by the torsion equa- 1 tion, the right hand side of (14) can be rewritten as δwab =0, δea = ε¯Γaψ, δψ =Dε. (20) 2 Dµ(ε¯Γaη)=Lξeaµ+ξνwbaνebµ−ξνψνΓaψµ. (15) The fields ea, wab and ψ transform as components of a connection for the super Poincaré group and therefore The first term in the right hand side of (15) represents a the supersymmetry algebra implied by (18)-(20) is the diffeomorphism with parameter ξµ =ε¯Γµη (Γµea =Γa), µ super PoincaréLie algebra. The invariance of the action the secondtermis arotationwithparameterξµwa , and bµ (17) under the super Poincaré group should come as no the third term is a supersymmetry transformation with surprise because it is the Chern-Simons action for the parameter −ξµψ . Equation (15) shows that, as far as µ connection A = eaP + 1wabJ + Q¯ψ, whose genera- the tetrad is concerned, the algebra of supersymmetry a 2 ab tors are P ,J and Qα. Indeed, the action (17) can be transformations closes. But the fact that we have used a ab written as [4] the torsion equation implies that the connection is no longeranindependentvariable. Onthecontrary,itsvari- 2 ation is given in terms of δea and δψ, and differs from I = <A∧dA+ A∧A∧A>, (21) Z 3 the form dictated by group theory. As a consequence, the local supersymmetry algebra acting on the gravitino where the bracket <···> stands for a properly normal- closes only on shell and auxiliary fields are required for ized trace on the algebra, with < J ,P >= ǫ and ab c abc its closure off shell. < Q ,Q >= −iC are the only non-vanishing traces. α β αβ Acompletelydifferentsituationisobservedinthecase [C is the charge conjugation matrix.] αβ of 2+1 gravity. Although the above discussion applies 3 III. FIVE DIMENSIONAL POINCARE´ (22) is also invariant under supersymmetry transforma- SUPERGRAVITY tions, δea =−i(ε¯Γaψ−ψ¯Γaε), We turn now to the supersymmetric versionof the action (12) in higher dimensions. To illustrate the ideas δwab =0, we start with the five dimensional case which already δb=ε¯ψ−ψ¯ε, contains the main ingredients. The general case will be δψ =Dε. (25) indicated in the next section. The proof of the invariance of (22) under (25) is straightforward. An important test of the consistency A. The Action of (25) is the fact that the commutator of two supersymmetry transformations gives local translation (24). The analog of the action (17) in five dimensions has Indeed, if δ and δ are supersymmetry transformations ε η three pieces; with parameters ε and η, we have, I5/susy =IG+Ib+Iψ (22) [δε,δη]ea =−iD(ε¯Γaη−η¯Γaε), [δ ,δ ]wab =0, where I is the purely gravitationalterm, I is a second ε η G b bosonic term needed by supersymmetry, and Iψ is the [δε,δη]b=d(ε¯η−η¯ε), fermionic term. The explicit formulas are: [δ ,δ ]ψ =0. (26) ε η 1 IG = 2Z ǫabcdfRab∧Rcd∧ef, locTahlesusypmerm-Peotirniecsaoréftgheenaecratitoonrs(2P2),arJeg,enQeαra,teQ¯db,yptlhues a ab α the Abelian generator K, responsible for the non-zero Ib =Z Rab∧Rab∧b, transformation of b in (24). These generators form an extension of the super-Poincaréalgebrawhose only non- Iψ =−Z Rab∧(ψ¯Γab∧Dψ+Dψ¯Γab∧ψ). (23) vanishing (anti-)commmutator is Here the gravitino field is a Dirac spinor 1-form, and b {Qα,Q¯β}=−i(Γa)αβPa+δβαK, (27) is a 1-form Lorentz pseudo-scalar. The form of I is dic- b plus the Poincaré algebra. The commutators of K, Qα, tated by the transformation of I and I under super- G ψ andQ¯ withtheLorentzgeneratorscanbe readofffrom symmetry. [See section V for an alternative way to see α their tensor character. this through the integrability conditions of the classical Note that K commutes with all the generators in the equations of motion.] algebra and therefore, it is as a central charge in the su- Since we are interested in the geometrical aspects per Poincaré algebra. This is, however, a peculiarity of of this theory only, we have set all the coupling con- five dimensions. For other odd dimensions, the genera- stants equal to one. The Lorentz covariant derivative tor K is a completely antisymmetric tensorof fifth rank, D in the spinorial representation is given by Dψ = dψ + 14wabΓab∧ψ. We work with Dirac spinors in or- wgehniecrhahtoars.a non-vanishingcommutatorwith the Lorentz der to avoid the dimensional dependence of Majorana spinors which only exist -on a Minkwoskian signature- in dimensions 2,3,4 mod 8. All the main results of this B. Chern-Simons formulation paper carrythroughforMajoranaspinorswhen they exist. The Dirac conjugate is defined as ψ¯ = ψ†Γ and in 0 the action, ψ and ψ¯ are varied independently. [See the The fact that the symmetries of the action (22) close Appendix.] without the need of any auxiliary fields strongly suggest Besides localLorentzrotations,the aboveactionis in- that (22) may be written as a Chern-Simons action for variant under the Abelian translations, the supergroup. In this section we prove that this is indeed the case. δea =Dλa, ConsideraconnectionAforthe superalgebrafoundin δwab =0, the last section, δb=dρ, 1 A=eaP + wabJ +bK+ψ¯ Qα−Q¯ ψα. (28) δψ =0, (24) a 2 ab α α where λa is a 0-form Lorentz vector and ρ is a 0-form The super-curvature F = dA + A∧A is then found by direct application of (5) and (27) Lorentzpseudo-scalar. Theinvarianceof(22)under(24) follows directly from the Bianchi identity. The action 4 F =FAG Justasinthe fivedimensionalcase,the supersymmet- A 1 ric action in dimension 2n+1 has three terms; =TãP + RabJ +F˜K+Dψ¯ Qα−Q¯ Dψα. (29) a ab α α 2 I =I +I +I (34) 2n+1/susy G b ψ Here Tã := Ta−iψ¯Γa∧ψ and F˜ := db+ψ¯∧ψ; Ta is the where the bosonic ‘geometric’ term I is given by, torsion2-form, and Rab is the 2-form Lorentz curvature. G We recall that a Chern-Simons Lagrangian in five dimensions, Lg, is defined by the relation IG =Z Rabc∧Rab∧ec. (35) gABCFA∧FB∧FC =dLg, (30) Ib is a second bosonic term involving a fifth-rank 1-form field, babcde, where the trilinear form g ≡< G ,G ,G > is an ABC A B C 1 DinivffaerrieannttctheoniscoersoofftthheeiLnivearailagnetbtreanwsoitrhggenergaivtoerdsiffGeAr-. Ib =−6Z Rabc∧Rde∧babcde. (36) ABC ent five dimensional Lagrangians L . g (Note that this term vanishes in three dimensions and To prove that (22) is a Chern-Simons action we need in five dimensions babcde is a Lorentz pseudo-scalar.) Fi- to find aninvarianttensor such that L is equal, up to a g nally, the fermionic part is total derivative, to the Lagrangian in (22). This tensor indeed exists and is given by, i Iψ = 3Z Rabc∧(ψ¯Γabc∧Dψ+Dψ¯Γabc∧ψ). (37) <J ,J ,P > =ǫ , ab cd e abcde Each term in the action (34) is independently invariant <J ,J ,K > =η η −η η ab cd ac bd ad bc under local Lorentz transformations. The complete ac- <Qα,J ,Q¯ > =−2(Γ )α. (31) ab β ab β tion is invariant under the Abelian translations, Itisstraightforwardtoprovethat,uptoatotalderivative δea =Dλa, δwab =0, δbabcde =Dρabcde, and an overall factor, the 5-form L associated to the g δψ =0, (38) above tensor is equal to the supersymmetric Lagrangian in (22). and supersymmetry transformations, ¿From the above result it is now evident that the action is invariant under supersymmetry transformations δea =−i(ε¯Γaψ−ψ¯Γaε), δwab =0, up to a total derivative. All the symmetry transforma- δbabcde =−i(ε¯Γabcdeψ−h.c.), δψ =Dε, (39) tions of the action can now be collected together in the formδA=∇λwhere ∇is the covariantderivativeofthe where D represents the Lorentz covariant derivative. supergroup and λ is a zero-form Lie algebra-valued vec- The proof of the invariance of (34) under supersym- tor in the adjoint representation. From this formula it is metry transformations is straighforward. One starts by alsoevidentthatthe algebraofsupersymmetrytransfor- varying the fermionic part. Up to a boundary term one mations closes as dictated by group theory, easily obtains i [δλ,δη]A=δ[λ,η]A. (32) δIψ = 12Z Rabc∧Rde∧(ψ¯{Γabc,Γde}ǫ−c.c.). (40) The Chern-Simons action (22) can be obtained from the Using the formula (A4) of the Appendix, we find that action found in [5] by an appropriateWigner-Inonucon- (40) has a term proportional to a 5-rank Dirac matrix traction. The closure of the supersymmetry algebra, Γabcde plus a term proportional to a Dirac matrix Γa. however,wasnotmentionedin[5]. Inthenextsectionwe It is direct to see that the first term is cancelled by the provethat the aboveschemeis notexclusiveofthe three variation of I while the second term is cancelled by the and five dimensional theories but it can be extended to b variation of I . any odd-dimensional spacetime. G As inthe lowerdimensionalcases(D =3,5),the commutator of two supersymmetry transformations gives a localAbeliantranslation(38). Thus,thesupersymmetric IV. THE GENERAL CASE extension of the Poincaré algebra that leaves the action invariant has generators G = [P ,J ,K ,Qα,Q¯ ]. A a ab abcde α Inthissectionweshowhowtheresultsofthe previous The only non-vanishing (anti-)commutator is sectionsaregeneralizedtoanyodd-dimensionalmanifold. Inordertosimplifythenotationweintroducethesymbol {Qα,Q¯ }=−i(Γa)αP −i(Γabcde)αK , (41) β β a β abcde R defined by, abc plus the Poincaréalgebra. The commutatorsofthe Q,Q¯ Rabc :=ǫabca1···aD−3Ra1a2∧···∧RaD−2aD−3. (33) and K with the Poincaré generators can be read from their tensorial character. 5 The action I is also a Chern-Simons action. VI. COMMENTS AND PROSPECTS 2n+1/susy The connection now is We have shown in this note that the successful meth- 1 A=eaPa+ wabJab+babcdeKabcde+ψ¯αQα−Q¯αψα, ods used in three dimensions to construct supersymet- 2 ric extensions of general relativity can be generalized to (42) any odd-dimensional spacetime. We have restricted our- selves, however, to Poincaré supergravity. The full anti- andtheLagrangianisdefinedby<F∧···∧F >=dL2n+1 de Sitter extension remains an open problem. [In five where the invariant (n+1) multilinear form < ··· > is dimensions, a Chern-Simons action for anti-de Sitter su- defined by pergravityhasbeenknownforsometime[5]. Thataction reduces to the action consideredhere after a proper con- <J ···J P > =ǫ , a1a2 aD−2aD−1 aD a1···aD traction is performed.] 1 There are good reasons to seek a full anti-de Sitter <J ···J K > =− ǫ η , a1a2 fg abcde 12 a1···aD−3abc [fg][de] Chern-Simons formulation of supergravity. First, the <QJa1a2···JaD−4aD−3Q¯ > =2inΓa1···aD−3 (43) bosonic Lagrangian in the Poincaré case does not con- tain the Hilbert term thus making the contact with four (the remaining brackets are zero). This completes the dimensional theories rather obscure [6]. Secondly, the construction of the (2n+1)-dimensional Chern-Simons Poincarétheoryinodddimensionsdoesnotpossessblack action for supergravity. hole solutions while the anti-de Sitter theory does [7]. In principle, a Chern-Simons anti-de Sitter supergravity can be constructed from the knowledge of the as- V. INTEGRABILITY OF THE EQUATIONS OF sociated supergroup and an invariant tensor only (find- MOTION. ing the invariant tensor, however, may prove to be a non-trivial task). In five dimensions, the relevant super- A remarkable feature of supersymmetric theories in groupisSU(2,2|1)[17]whileintheimportantexampleof generalandsupergravityinparticularisthefactthatthe eleven dimensions the supergroup is OSp(32|1) [18]. As integrability conditions for the fermionic field equations the spacetime dimension increases, one faces a growing arethe bosonic equations. The study ofthe integrability multiplicity of choices for the invariant tensor. To illus- conditions of the fermionic equation in our model sheds trate this issue, consider the problem of classifying all some light on the role of the bosonic field babcde. the invariants that can be constructed out the Lorentz In the notation introduced in Sec. IV, the fermionic curvature in a given dimension [19]. In four dimensions field equations are we only have, RabcΓabc∧Dψ =0, (44) ǫabcdRab∧Rcd, Rab∧Rab, (48) while in eight dimensions we have, and similarly for the Dirac conjugate spinor. Taking the covariantderivative of(44)we find the integrabilitycon- ǫabcdefghRab∧Rcd∧Ref∧Rgh, Rab∧Rab∧Rcd∧Rcd, dition, Rab∧Rbc∧Rcd∧Rda. (49) Rabc∧RdeΓabcΓde∧ψ =0. (45) Of course,all the above scalarsdefine Chern-Simons La- grangians in dimension five and seven respectively. A This equation should be satisfied for any ψ. Using ele- similar proliferation of scalars appears in supergravity. mentary properties of the Dirac matrices we obtain the A good candidate for the right theory could be a linear following equations for the bosonic fields, combinationofallpossibleinvariantssuchthat,underan appropriated Wigner-Inonu contraction, reduces to the Rabc∧RabΓc =0, (46) Poincarétheories studied here. Rabc∧RdeΓabcde =0. (47) The particular case of eleven dimensions seems to be particulary suited to admit an anti-de Sitter Chern- Eq. (46) is the equation of motion for the vielbein field, Simons formulation. As shown in [18], the super anti- whileEq. (47)istheequationofmotionfortheb−field. de Sitter group is OSp(32|1). A natural basis for the Had we not included b, supersymmetry would not have Lie algebra of Sp(32) is given by the Dirac matrices beenachievedandthe integrabilityconditionswouldnot Γ ,Γ ,Γ ,andthisbasisiseasilyextendedtoexpand a ab abcde havebeensatisfied. [ThisdoesnotruleoutaLagrangian the superalgebra of OSp(32|1). Thus, the supergroup withoutthe bfield. However,the fermionic equationsfor OSp(32|1) naturally accommodates the field content of such a theory would impose additional equations on the thePoincaréChern-Simonssupergravityconsideredhere. bosonic fields.] One could expect, therefore, that a Chern-Simons La- grangianfor the supergroup OSp(32|1) in eleven dimensions should reduce to the supersymmetric action (34) 6 uponcontraction. For example,is easyto check thatthe We define Γa1···ap as the totally antisymmetric product superalgebraobtainedinSec. (IV) canbe obtainedfrom of gamma matrices, OSp(32|1) by a Wigner-Inonu contraction: 1 Γa1···ap = sgn(σ)Γa ···Γa , (A2) Ga→λ−1Pa, p!Xσ σ(1) σ(p) G →J , ab ab Two useful formulas implicitly used in the text are: G →λ−1K , abcde abcde Qα→λ−12Qα. (50) Γ =−(−i)n+1ǫ Γabc (A3) a1···aD−3 3! a1···aD−3abc At the level of the Lagrangian, however, the problem is more complicated. Due to the ambiguity in the choice 1 of the invariant tensors and the large number of terms {Γabc,Γde}=Γabcde−[η[ab][de]Γc+perm(abc)], (A4) 2 in the super anti-de Sitter Chern-Simons action, it is a non-trivial problem to find an expression such that, un- where η[ab][de] =ηadηbe−ηaeηbd. der contraction, reduces to the action considered in this paper [20]. Finally we mention that the -Poincaré- supersymmet- Acknowledgments ric Chern-Simonsactionsfound inthis paper arenotthe only possibilities. In dimensions greater than five, the TheauthorsaregratefultoM.Contreras,C.Mart´ınez fermionic Lagrangian accepts other Poincaré-invariant and, especially, C. Teitelboim for helpful discussions. terms that give rise to other supergravities. For exam- This work was supported in part by grants 1930910and ple, in eleven dimensions one can add to the fermionic 2950047 from FONDECYT (Chile), 27-953/ZI-DICYT lagrangianthe term, (University of Santiago) and grant No. PG-023-95 of Departamento de Postgrado y Post´ıtulo (University of [Rab∧Rab]2∧ψ¯∧Dψ (51) Chile). M.B. was partially supported by a grant from Fundación Andes. R.T. thanks CONICYT for financial This term, however, requires extra bosonic fields to re- support. Institutional support to CECS from a group spect supersymmetry. This is easily seeing by studying of Chilean private companies (COPEC, CGEI, Empre- the integrabilityconditions generatedby (51). One finds sasCMPC,ENERSIS,MINERALAESCONDIDA,IBM the equation over the bosonic fields, and XEROX-Chile) is also acknowledged. [Rab∧Rab]2∧Rcd =0. (52) Thus, consistency requires an extra bosonic term in the Lagrangian of the form [Rcd∧Rcd]2∧Rab∧cab which in- volves the 1-form c . Varying this term with respect ab to c gives (52). Thus, the integrability condition is ab satisfied and the action is supersymmetric. A complete [1] Foracompletetreatmentsee,M.B.Green,J.H.Schwarz classification of all possible fermionic Lagrangians for a and E. Witten, Superstrings, Cambridge University givendimensionandtheircorrespondingsupersymmetry Press, 1987. [2] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76, algebrasis beyondthe scope ofthis work. We wouldlike 409 (1978). to point out, however, that the method outlined here [3] E. Witten,Nucl. Phys. B 311, 46 (1988). seems to provide a simple way to generate extensions of [4] A. Achućarro and P.K. Townsend, Phys. Lett. B180, 89 thesuper-Poincaréalgebrainvolvingextrabosonicfields. (1986). [5] A.H. Chamseddine, Nucl. Phys. B 346, 213 (1990). [6] A.H. Chamseddine, Phys. Lett. B233, 291 (1989). APPENDIX A: GAMMA MATRICES [7] M. Banãdos, C. Teitelboim and J.Zanelli, Phys. Rev. D49, 795 (1994). See also, JJ Giambiagi Festschriff, H. The Clifford algebra in D = 2n+1 dimensions with Falomir, R. Gamboa, P. Leal and F. Shaposnik eds. Minkowskian signature can be generated by a set of (World Scientific, Singapore, 1991). 2n×2n matrices; the unit I and D matrices Γa, satisfy- [8] M.Banãdos,C.TeitelboimandJ.Zanelli,Phys.Rev.Lett ing {Γa,Γb}=2ηabI, where a,b,...,= 1,2,...,2n+1 and 69, 1849 (1992). ηab=diag(−,+,···,+). In this signature, Γ† =Γ Γ Γ . [9] M.Banãdos,L.J.GarayandM.Henneaux,Phys.Rev.D, a 0 a 0 It is alwayspossible to finda representationofΓa main press. trices in which [10] T. Regge, Phys. Rep. 137, 31 (1986). [11] D. Lovelock, J. Math. Phys. 12, 498 (1971). Γ:=Γ1Γ2···ΓD =(−i)n+1I. (A1) [12] S. Deser, R. Jackiw and G. ’t Hooft, Ann. Phys. 152, 220 (1984). 7 [13] M. Banãdos, L.J. Garay and M. Henneaux, in preparation. [14] C.Teitelboim,Phys. Rev. Lett.38,1106(1977);C.Teit- elboim and R.Tabensky,Phys. Lett. 69 B, 240 (1977). [15] ForareviewseeP.vanNieuwenhuizen,Phys. Rep.68,4 (1981). [16] ForearlierworkonthreedimensionalsupergravityseeN. MarcusandJ.H.Schwarz,Nucl.Phys.B228,145(1983); S.Deser and J.H. Kay,Phys. Lett. B120, 97 (1983). [17] W. Nahm,Nucl. Phys. B135, 149 (1978). [18] J. W. van Holten and A. Van Proeyen, J. Phys. A15, 3763 (1982). [19] A. Mardones and J. Zanelli, Class. & Quan. Grav. 8, 1545 (1991). [20] M. Banãdos, R. Troncoso and J. Zanelli, in preparation. 8